New proof for the undecidability of the circular PCP

We give simpler proof for the undecidability of the circular Post Correspondence Problem that was originally proved undecidable by Ruohonen (Acta Informatica 19:357–367, 1983). The key feature of our proof is the undecidability of the word problem of special semi-Thue systems where derivations are deterministic and reversible for words containing a single occurrence of a letter from a given set.     

On decidability and closure properties of language classes with respect to bio-operations

We present general results that are useful in showing closure and decidable properties of large classes of languages with respect to biologically-inspired operations. We use these results to prove new decidability results and closure properties of some classes of languages under bio-operations such hairpin-inversion, the recently studied operation of pseudo-inversion, and other bio-operations. We also provide techniques for proving undecidability results. In particular, we give a new approach for proving the undecidability of problems for which the usual method of reduction to the undecidability of the Post Correspondence Problem seems hard to apply. Our closure and decidability results strengthen or generalize previous results.     

Extension of the decidability of the marked PCP to instances with unique blocks

In the Post Correspondence Problem (PCP) an instance (h,g)$(h,g)$ consists of two morphisms h$h$ and g$g$, and the problem is to determine whether or not there exists a nonempty word w$w$ such that h(w)=g(w)$h\left(w\right)=g\left(w\right)$. Here we prove that the PCP is decidable for instances with unique blocks using the decidability of the marked PCP. Also, we show that it is decidable whether an instance satisfying the uniqueness condition for continuations has an infinite solution. These results establish a new and larger class of decidable instances of the PCP, including the class of marked instances.     

On simplest possible solutions for Post Correspondence Problems

We investigate the simplicity of solutions for instances of the Post Correspondence Problem, from the point of view of both index words and terminal words. This leads to the notion of amixed primality type of an instance. Our main result gives an exhaustive characterization of mixed primality types.     

Undecidability of state complexity

We consider the state complexity of compositions of regularity-preserving language operations. As our main result, we establish that determining the state complexity of an operation composed just from intersections and marked concatenations is undecidable. The proof relies on the undecidability of Hilbert's Tenth Problem. The languages used in the construction are over a fixed alphabet with at most 50 letters. We also consider some implications and generalizations, as well as discuss open problems.     

On the membership of invertible diagonal and scalar matrices

In this paper, we consider decidability questions that are related to the membership problem in matrix semigroups. In particular, we consider the membership of a given invertible diagonal matrix in a matrix semigroup and then a scalar matrix, which has a separate geometric interpretation. Both problems have been open for any dimensions and are shown to be undecidable in dimension 4 with integral matrices by a reduction of the Post Correspondence Problem (PCP). Although the idea of PCP reduction is standard for such problems, we suggest a new coding technique to cover the case of diagonal matrices.     

P, NP, and the Post Correspondence Problem

We define a variant of the Post Correspondence Problem, the machine-oriented Post Correspondence Problem or MOPCP, especially suitable for complexity considerations. All recursively enumerable sets can be represented in terms of instances of MOPCP and, moreover, deterministic and nondeterministic time and space complexities have their natural counterpart in the representation. This leads to PCP-related complexity classes; for instance, the time complexity classes PCP-P and PCP-NP. Using a bounded delay injectivity condition on one of the morphisms of a PCP instance, we obtain an exact characterization of P. In this characterization, determinism corresponds exactly to the bounded delay condition and no reference is made to computations of any sort. A weaker bounded delay requirement gives rise to an infinite (possibly collapsing) hierarchy between P and NP.     

The Structure of Infinite Solutions of Marked and Binary Post Correspondence Problems

In the infinite Post Correspondence Problem an instance (h,g) consists of two morphisms h and g, and the problem is to determine whether or not there exists an infinite word ω such that h(ω) = g(ω). This problem is undecidable in general, but it is known to be decidable for binary and marked instances. A morphism is binary if the domain alphabet is of size 2, and marked if each image of a letter begins with a different letter. We prove that the solutions of a marked instance form a set E^ω ⋃ E^* (P ⋃ F), where P is a finite set of ultimately periodic words, E is a finite set of solutions of the PCP, and F is a finite set of morphic images of fixed points of D0L systems. We also establish the structure of infinite solutions of the binary PCP.     

Robust parametric reachability for timed automata

We review the known decidability and undecidability results for reachability in parametric timed automata. Then, we present a new proof of undecidability in dense time for open timed automata that avoids equalities in clock constraints. Our result shows that the undecidability of parametric timed automata does not follow from their ability to specify punctual constraints in a dense time domain.     

Recognizability of the support of recognizable series over the semiring of the integers is undecidable

A recognizable series over the semiring of the integers is a function that maps each word over an alphabet to an integer. The support of such a series consists of all those words which are not mapped to zero. It is long known that there are recognizable series whose support is not a recognizable, but a context-free language. However, the problem of deciding whether the support of a given recognizable series is recognizable was never considered. Here we show that this problem is undecidable. The proof relies on an encoding of an instance of Post?s correspondence problem.     

On the cardinal number of complete sets of Boolean operations†

It is shown in the literature (using the Post—Yablonsky theorem) that a complete set of Boolean operations cannot have a cardinal number greater than four. It is the object of this paper to improve this bound and prove that a complete set can have a cardinal number of at most three or, in other words, there does not exist a complete (non–redundant) set of more than three Boolean operations. The proof given here is constructive, using the Post—Yablonsky theorem, truth tables and combinatorial set theory.     

Decision problems for semi-Thue systems with a few rules

We show that the accessibility problem, the common descendant problem, the termination problem and the uniform termination problem are undecidable for 3-rules semi-Thue systems. As a corollary we obtain the undecidability of the Post correspondence problem for 7 rules.     

The periodic domino problem revisited

In this article, we give a new proof of the undecidability of the periodic domino problem. Compared to previous proofs, the main difference is that this one does not start from a proof of the undecidability of the (general) domino problem but only from the existence of an aperiodic tileset.     

A Branch-and-Prune algorithm for the Molecular Distance Geometry Problem

The Molecular Distance Geometry Problem consists in finding the positions in of the atoms of a molecule, given some of the inter‐atomic distances. We show that under an additional requirement on the given distances this can be transformed to a combinatorial problem. We propose a Branch‐and‐Prune algorithm for the solution of this problem and report on very promising computational results.     

化工模拟系统中面向问题语言的结构和翻译

面向问题语言（ＰＯＬ）是大型化工模拟软件不可缺少的组成部分，本文描述了一种统一的ＰＯＬ结构，从而使得翻译ＰＯＬ的程序具有通用性，这一通用性保证了任何符合ＰＯＬ语法结构的语包可以由ＰＯＬ翻译程序进行翻译，而不必修改任何程序。     

Bounds for the Element Distinctness Problem on one-tape Turing machines

We disprove a conjecture of López-Ortiz by showing that the Element Distinctness Problem for n numbers of size O(logn) can be solved in O(n²(logn)^3/2(loglogn)^1/2) steps by a nondeterministic one-tape Turing machine. Further we give a simplified algorithm for solving the problem for shorter numbers in time O(n²logn) on a deterministic one-tape Turing machine and a new proof of the matching lower bound.     

An adaptive large neighborhood search heuristic for Two-Echelon Vehicle Routing Problems arising in city logistics

In this paper, we propose an adaptive large neighborhood search heuristic for the Two-Echelon Vehicle Routing Problem (2E-VRP) and the Location Routing Problem (LRP). The 2E-VRP arises in two-level transportation systems such as those encountered in the context of city logistics. In such systems, freight arrives at a major terminal and is shipped through intermediate satellite facilities to the final customers. The LRP can be seen as a special case of the 2E-VRP in which vehicle routing is performed only at the second level. We have developed new neighborhood search operators by exploiting the structure of the two problem classes considered and have also adapted existing operators from the literature. The operators are used in a hierarchical scheme reflecting the multi-level nature of the problem. Computational experiments conducted on several sets of instances from the literature show that our algorithm outperforms existing solution methods for the 2E-VRP and achieves excellent results on the LRP.     

TM图象对应分析研究

利用对应分析法对对内蒙色尔腾山地区的ＴＭ图象进行了分析,达到了图象分类和增强特征信息的目的,同时列出了它所提供的有用信息。     

Decidability and complexity of Petri nets with unordered data

We prove several decidability and undecidability results for ν-PN, an extension of P/T nets with pure name creation and name management. We give a simple proof of undecidability of reachability, by reducing reachability in nets with inhibitor arcs to it. Thus, the expressive power of ν-PN strictly surpasses that of P/T nets. We encode ν-PN into Petri Data Nets, so that coverability, termination and boundedness are decidable. Moreover, we obtain Ackermann-hardness results for all our decidable decision problems. Then we consider two properties, width-boundedness and depth-boundedness, that factorize boundedness. Width-boundedness has already been proven to be decidable. Here we prove that its complexity is also non-primitive recursive. Then we prove undecidability of depth-boundedness. Finally, we prove that the corresponding “place version” of all the boundedness problems is undecidable for ν-PN. These results carry over to Petri Data Nets.